Write the balanced chemical equations for the following reactions:
$(1)$ Reaction between $MnO_4^-$ and $I^-$ in neutral or faintly alkaline solution.
$(2)$ Reaction between $MnO_4^-$ and $C_2O_4^{2-}$ in acidic medium $(H^+)$.

Balance the following redox reactions by the ion-electron method:
$(a)$ $MnO_{4(aq)}^- + I_{(aq)}^- \to MnO_{2(s)} + I_{2(s)}$ (in basic solution)
$(b)$ $MnO_{4(aq)}^- + SO_{2(g)} \to Mn_{(aq)}^{2+} + HSO_{4(aq)}^-$ (in acidic solution)
$(c)$ $H_2O_{2(aq)} + Fe_{(aq)}^{2+} \to Fe_{(aq)}^{3+} + H_2O_{(l)}$ (in acidic solution)
$(d)$ $Cr_2O_7^{2-} + SO_{2(g)} \to Cr_{(aq)}^{3+} + SO_{4(aq)}^{2-}$ (in acidic solution)

Molecular weight of $Cu_2S$ is $M$. In the following reaction its equivalent weight is $M / x$. What is $x$?
$Cu_2S + MnO_4^{-} \rightarrow Cu^{2+} + Mn^{2+} + SO_2$

Manganese ions $(Mn^{2+})$ can be oxidised by persulphate ions $(S_2O_8^{2-})$ according to the following half-equations:
$S_2O_8^{2-} + 2e^- \longrightarrow 2SO_4^{2-}$
$Mn^{2+} + 4H_2O \longrightarrow MnO_4^- + 8H^+ + 5e^-$
How many moles of $S_2O_8^{2-}$ are required to oxidise $1 \ mole$ of $Mn^{2+}$?

The value of $x$ in the partial redox equation $MnO_4^- + 8H^{+} + xe^- \rightleftharpoons Mn^{2+} + 4H_2O$ is

Number of moles of $MnO_4^-$ required to oxidize one mole of ferrous oxalate completely in acidic medium will be .......... $moles$.